Computational Materials Science

Although categorized as an engineering field which normally studies artificial materials, metallurgy has also treated natural objects in its long history. Metallurgy and Alchemy, which are the basis of present-day materials science, have also produced all the natural sciences, including chemistry and physics. The field of materials science is expanding rapidly to include amorphous, ceramic, polymer, and nanoscale materials.

In fields such as materials science, electronics, mechanical engineering and bioscience, not to mention physics and chemistry, the keywords “first principles” and “ab initio” have been widely used recently. The principal idea of these keywords is to regard many-atom systems as many-body systems composed of electrons and nuclei, and to treat everything on the basis of first principles of quantum mechanics, without introducing any empirical parameters.

Despite recent major developments in algorithms and computer hardware, the simulation of large systems of particles by ab initio methods is still limited to about a hundred particles. For treating larger systems by molecular dynamics, one can use either tight-binding (TB) or classical molecular- dynamics methods. The TB method has the advantage of being quantum mechanical; therefore one has, in addition to its higher accuracy, information about the electronic structure of the system. In the field of quantum chemistry, other semi-empirical methods, such as MNDO (modified neglect of differential overlap), also exist. These are, in their nature, very similar to Hartree–Fock methods, but the computations of the Hamiltonian and overlap matrix elements are based on semi-empirical formulae.

In the previous two chapters, we learned that atomistic- and electronic-scale simulations can be performed by means of ab initio methods or semi-empirical methods such as a tight-binding method. However, we learned also that these are, at present, still restricted in their capability with respect to both the number of atoms and the simulation timescale. In order to study longer-timescale phenomena of systems composed of larger numbers of particles, it becomes necessary to introduce much easier but still atomic scale methods. If such methods are more or less based on the ab initio or semi-empirical total energies and they are not simplified too much, one may rely on these methods as a substitute. The important issue here is how to reduce the amount of necessary computation in such methods, and how to introduce parametrizations or fittings into the interatomic potentials without losing too much accuracy and reliability. In principle, it is possible to construct realistic classical potentials based on ab initio calculations. A possible methodology here is to determine classical potentials by, for example, fitting them to contour maps of the total energy, which may be obtained with an ab initio method by changing the position of one atom while fixing the coordinates of all other atoms. In this book, we do not go into details of the applications of classical molecular dynamics, since readers can refer to many good reviews. Instead, a few examples are given of how classical potentials can be constructed from ab initio theories.

Nature is composed of gross assemblies of huge numbers of atoms and molecules showing a wide variety of phenomena according to the way how they are assembling. The macroscopic behaviors of such systems are rather different from the microscopic laws in the world of atoms and molecules. For example, in the usual cases of macroscopic systems, the motion of the atoms and molecules can be regarded simply as heat. That is, the average kinetic energy of each atom and molecule in a macroscopic system is equal to a quantity measured as the temperature. The other contributor to macroscopic behavior is the cooperative motion of atoms and molecules (or sometimes electrons). Since atoms and molecules interact with each other, their macroscopic assemblies can have cooperative motions. Many examples can be seen in our daily life: spring or rubber elasticity, magnetization in permanent magnets, shape memory alloys, liquid flows, surface tension of liquids, swelling of polymers by water absorption, etc. Sometimes such cooperative motions are frozen as the temperature decreases. In this case, the states which have been realized at higher temperatures become unstable.

Since Metropolis’s work , the Monte Carlo method has been applied not only to various statistical problems of classical systems, but also to many quantum mechanical systems. In this chapter, we briefly describe the extension of the Monte Carlo method to quantum mechanical systems.